Example 3.3 [1]}
Let \(R=Q[[x,y]]\)  and \(I=(x^4,x^3y,xy^3,y^4)\) . Then \(x^2y^2\in I^2:I \subseteq \widetilde{I}\)  but \(x^2y^2\notin I\)  which implies \(\operatorname{depth}G(I)=0\) . Note that \(J=(x^4,y^4)\)  is a minimal reduction of \(I\)  and \(p=x^4+y^4\)  is superficial for \(I\)  as \(e_0(I)=16=e_0(I/(p))\)  and \(e_1(I)=6=e_1(I/(p))\) . Further,
\(r_J(I)=2=r_{J/(p)}(I/(p)).\)

